The 'centre of gravity' or 'centre of mass' of any collection of objects (with mass!) can - in principle - be calculated with some simple math; it's just a point. However, it's a very convenient 'point'! Re-writing your (Newtonan) equations of motion using this point as the origin of your coordinate system makes the math easier to crunch.

Let's assume that your favourite globular cluster is spherical and has no net angular momentum (some globulars aren't like this; e.g. Omega Cen - the brightest as seen from Earth - is clearly somewhat elliptical). Then its 'centre' will be the centre of mass (assuming we define 'centre' appropriately). What lies at the centre of a globular cluster? In most cases, nothing special. Unlike elliptical (or spiral) galaxies, there is no obvious nucleus in globulars ... a curve of light intensity (for example) vs radius is smooth with no 'spike'. This means that the density of stars - say, per cubic light-year - increases smoothly to a maximum. Further, the motions of stars in the inner part of a globular suggest that there isn't a supermassive black hole lurking at the centre.